The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this convergence on all mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum with an optimal convergence rate. As a corollary we obtain complete delocalisation for the corresponding eigenvectors in any basis.
The first author gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 895698, from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715539 RandMat) and from the Swiss National Science Foundation through the NCCR SwissMAP grant.
The second author gratefully acknowledges financial support from Novo Nordisk Fonden Project Grant 0064428 & VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and Young Investigator Award (Grant No. 29369).
"Local elliptic law." Bernoulli 28 (2) 886 - 909, May 2022. https://doi.org/10.3150/21-BEJ1370